Decoupling of the fluid viscosity and incompressibility constraint in a partitioned scheme

I am new here and this is my very first question: I hope I respect all the criteria and rules.
I am just getting started with partitioned approaches for solving FSI problems. I am interested in a Chorin-Temam projection scheme for the fluid, with semi implicit coupling. This is the paper I am referring to: https://arxiv.org/abs/1711.10829

I am having some doubts regarding the following fact: if, for visualization purposes, I save the fluid velocity $$\hat{\textbf{u}_f}$$ that I compute in the explicit step, then what I am visualizing is not the physical velocity, correct? Because $$\hat{\textbf{u}_f}$$ does not satisfy the incompressibility constraint.

Sorry if this is an easy question, but I have just started to study partitioned approaches and would like to understand also the very basic aspects about them.

• As far as I see, they are lagging the solution (a way of linearizing the equations). If it converges, it should converge to the actual solution and eventually (after few time steps) $\hat{u}_f$ should satisfy the incompressibility constraint. However, I don't see under what conditions it would converge so there is a little bit doubt on that matter. – Abdullah Ali Sivas Mar 1 '20 at 2:49
• Also, the FEM they use only weakly satisfies the incompressibility constraint so if you are interested in pointwise divergence-free solution you should use a different finite element method. – Abdullah Ali Sivas Mar 1 '20 at 2:53
• Ok, so you are saying that, at the beginning of my simulation $\hat{\textbf{u}}_f$ does not satisfy the incompressibility constraint, but after a few time steps, it will? Yes, so far I have only been interested in weakly satisfying the incompressibility constraint. – user111283 Mar 2 '20 at 9:26
• Yes, pretty much that. Also, -necessary details are not in the article so I am not sure- but if $\hat{u}_f$ at time $t=0$ (weakly) satisfies the incompressibility constraint $\hat{u}_f$ at following times should too, since they seem to be projecting onto such a subspace. – Abdullah Ali Sivas Mar 2 '20 at 21:35